Mészáros, Alpár Richárd and Silva, Francisco J. (2015) 'A variational approach to second order mean field games with density constraints : the stationary case.', Journal de mathématiques pures et appliquées., 104 (6). pp. 1135-1159.
In this paper we study second order stationary Mean Field Game systems under density constraints on a bounded domain . We show the existence of weak solutions for power-like Hamiltonians with arbitrary order of growth. Our strategy is a variational one, i.e. we obtain the Mean Field Game system as the optimality condition of a convex optimization problem, which has a solution. When the Hamiltonian has a growth of order , the solution of the optimization problem is continuous which implies that the problem constraints are qualified. Using this fact and the computation of the subdifferential of a convex functional introduced by Benamou and Brenier (see ), we prove the existence of a solution of the MFG system. In the case where the Hamiltonian has a growth of order , the previous arguments do not apply and we prove the existence by means of an approximation argument.
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||https://doi.org/10.1016/j.matpur.2015.07.008|
|Publisher statement:||© 2015 This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/|
|Date accepted:||No date available|
|Date deposited:||28 February 2020|
|Date of first online publication:||01 July 2015|
|Date first made open access:||28 February 2020|
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